Separability in distant Jauch-type hybrid macrostates of a quantum and a classical system (Q1821324)

In order to elaborate further a modification [the author, ibid. 25, 863- 876 (1986)] of Jauch's resolution of the problem of quantum measurement theory [\textit{J. M. Jauch}, Helv. Phys. Acta 37, 293-316 (1964; Zbl 0143.230)], it is assumed that a quantum system (Q) plus a classical one (C) interacted in the past and are now in a correlated composite state \(\rho_{I,II}(\neq \rho_ I\otimes \rho_{II})\), and that the actually measurable Hermitian operators are of the form \[ A_{I,II}\equiv A\otimes \sum_{k\in K}b_ kQ_ k \] (A is any Hermitian operator for Q, the decomposition of the identity \(\sum_{k\in K}Q_ k=1\) is characteristic for C, \(K\leq \aleph_ 0,\) and \((\forall k\in K)\quad b_ k\in R).\) The Jauch-type equivalence relation \(\rho_{I,II}\sim \rho '_{I,II}\) defined by \((\forall A_{I,II})\) \[ Tr_{I,II}\rho_{I,II}A_{I,II} = Tr_{I,II}\rho'_{I,II}A_{I,II}) \] then leads to the Jauch-type macrostate (actually classes of microstates \(\rho_{I,II})\) for \(Q+C.\) On the other hand, it is shown that in any \(Q+Q\) microstate \(\rho_{I,II}\) the essence of quantum correlations are the conditional states \[ \rho_ I(Q_ k)\equiv p_ k^{-1}Tr_{II}(1\otimes Q_ k)\rho_{I,II}\equiv \rho_ k,\quad p_ k\equiv Tr_{I,II}(1\otimes Q_ k)\rho_{I,II}, \] and the reduced state \(\rho_{II}\equiv Tr_ I\rho_{I,II}. \) It is shown that the correlation entities \(\{(\rho_ k,p_ k): p_ k>0\), \(k\in K\}\) are complete sets of invariants for the macrostates of the \(Q+C\) system. It is proved that one can map isomorphically the \(\sigma\)-convex set of all macrostates (on the composite state space \(H_ I\otimes H_{II})\) onto that of the hybrid macrostates \((\rho_ k,p_ k)\) (actually \(\{(\rho_ k,p_ k): k\in K\}\), if \(p_ k=0\), also \(\rho_ k\equiv 0)\). Here \(p_ k\) is a classical discrete probability distribution on K that takes the place of \(\rho_{II}\) (valid in the \(Q+Q\) case), and for \(p_ k>0\), \(\rho_ k\) is the conditional state of Q under the condition that \(Q_ k\) occurs on C. This study throws indirectly new light on the important nonseparability in the \(Q+{\mathbb{Q}}\) case by contrasting it with separability in the \(Q+C\) case (that is essentially not different from the well-understood \(C+C\) case). In further research, the author applies the hybrid macrostate formalism developed in this work to a new, relative collapse interpretation of quantum mechanics, in which every quantum mechanical statement is observer dependent, and the latter is defined via some observable \(B_{II}=\sum_{k\in K}b_ kQ_ k\) (as above, but now understood broader than just determining a classical system C).